Heun Method For Stochastic Differential Equations With Piecewise Continuous Arguments

Stochastic differential equations with piecewise continuous arguments can describe hybrid systems, contain the properties of difference equations and differential equations.Moreover, these equations consider the influence of noise on models and have been widely used in environmental science, economic, and other fields. Because of the difficulty in getting exact solutions to such stochastic differential equations, it makes sense to study numerical methods and their properties. Aiming at solving stochastic differential equations with piecewise continuous arguments, this paper will constructs Heun method and study the convergence, T-stability, as well as mean-square stability of this method.Chapter 1 primarily introduces the appearance and development of stochastic differential equations, the unique advantages in describing uncertain phenomenons, the extensive use in practical problems. Then it recommends the contribution made by some scholars,including numerical methods used to solve equations and the properties of them. At last, it presents the content that will be studied. Chapter 2 constructs Heun method and obtains the numerical solution to stochastic differential equations with piecewise continuous arguments by applying Heun method to them. Then, it makes an analysis on the convergence of this method, and gets a result that the convergence order of Heun method is 1/2. Chapter 3 applies Heun method to the linear stochastic differential equations with piecewise continuous arguments, analyzes the T-stability of it and gives the sufficient conditions of mean-square stability in the end. Chapter 4 studies the mean-square stability of Heun method applied to the equations in chapter3 and gives the sufficient conditions of mean-square stability as well. Chapter 5 shows four numerical experimentations to verify the corresponding theoretical results got above. At last, we come to a conclusion that theoretical results are consistent with numerical results. Chapter 6 summarizes the whole article and puts forward some own views for the later research.